Download Electromagnetic force on a magnetic dipole inside

PHYSICS OF FLUIDS 25, 097102 (2013)
Electromagnetic force on a magnetic dipole inside
an annular pipe flow
Christiane Heinickea) and André Thessb)
Institute of Thermodynamics and Fluid Mechanics, Ilmenau University of Technology,
P.O. Box 100565, 98684 Ilmenau, Germany
(Received 10 May 2013; accepted 15 August 2013; published online 5 September 2013)
We present an illuminating example of electromagnetic flow measurement in liquid metals that is easy to analyze yet displays a remarkably good agreement with
laboratory experiments. Our system involves a small permanent magnet located inside an annular pipe carrying the flow of a liquid metal. We investigate the Lorentz
force acting upon the magnet using a combination of laboratory experiments with
liquid metal at room temperature and a simple analytical model. We demonstrate that
the measured Lorentz forces are in very good agreement with the predictions of our
model over a wide range of geometry parameters. By virtue of its simplicity and close
relationship to the well known “creeping magnet” classroom experiment, our system
can also serve as an educational tool for introductory courses in liquid metal magnetoC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4820239]
hydrodynamics. I. INTRODUCTION
The “creeping magnet” is an easy-to-reproduce educational experiment:1, 2 A permanent magnet
is dropped inside a vertical copper pipe whose diameter is only slightly larger than that of the magnet.
Shortly after its release the magnet reaches a steady velocity that is significantly less than the free fall
velocity, as if the magnet was creeping through a highly viscous fluid. The cause of the retardation are
eddy currents induced by the movement of the permanent magnet inside the electrically conducting
(nonmagnetic) pipe. These eddy currents generate a secondary magnetic field which counteracts the
field of the permanent magnet and thus slows down its fall.
The present paper is devoted to a fluid-dynamic derivative of the creeping magnet problem
sketched in Fig. 1(a). Consider a quiescent magnet surrounded by a liquid metal that flows upward in
the annular gap between two vertical pipes. For the same reason as in the creeping magnet problem
the moving liquid metal creates an electromagnetic force (Lorentz force) on the magnet that is
directed upwards. If the flow is sufficiently fast, the Lorentz force can even compensate the gravity
force and levitate the magnet (at least in principle). Our motivation for considering this problem is
twofold.
The first reason to investigate this problem is that it represents one of the rare examples of
a magnetohydrodynamic (MHD) flow that is amenable to simple analytic treatment. By virtue of
the axisymmetry no electric potential must be computed and the Lorentz force can be explicitly
represented as a function of all relevant parameters. Moreover, the analytic model turns out to be
in very good agreement with a laboratory experiment and the results can be compared with the
above-mentioned creeping magnet problem.
The second reason for studying this problem is its practical background sketched in Fig. 1(b).
The present work is a scientific by-product of the development of a novel velocity sensor for liquid
metals that is based on the principle of Lorentz force velocimetry (see Ref. 3 and references therein).
Although Lorentz force velocimetry has been successfully used to measure global quantities such as
a) [email protected]
b) [email protected]
1070-6631/2013/25(9)/097102/9/$30.00
25, 097102-1
C 2013 AIP Publishing LLC
097102-2
C. Heinicke and A. Thess
(a)
Phys. Fluids 25, 097102 (2013)
(b)
FIG. 1. (a) Sketch of the problem setup. A magnetic dipole is located inside an annular gap, through which an electrically
conducting but nonmagnetic fluid flows. Magnetization of the dipole is directed along the symmetry axis which coincides
with the flow direction. (b) Conceptual sketch of the probe used in the experiment. A small magnet is immersed into the metal
flow. Protected by a thin tube, the magnet interacts with the metal in its proximity. The force is detected with a deflection
element (not shown) to which the magnet is attached with a lever (thick solid line).
mass fluxes under industrial conditions in metallurgy,4, 5 its application for the measurement of local
velocities is still under development. Figure 1(b) shows a simplified sketch of such a sensor that is
supposed to measure the local velocity of a liquid metal flow and that is currently being developed
by the authors of the present work. The sensor consists of a permanent magnet enclosed in a housing
and connected to a force-measurement system (not shown) measuring the streamwise component of
the Lorentz force acting upon the magnet. From the measured Lorentz force the flow velocity in the
vicinity of the sensor tip can be deduced. The design, the directional characteristics, and the specific
applications of this velocity sensor are outside the scope of the present paper. However, we have
discovered that the sensor is well suited for an experimental realization of the idealized problem
sketched in Fig. 1(a).
Since the primary motivation of our work comes from the creeping magnet problem, we start
with formulating and solving an analytical model in Sec. II. More specifically, we determine the
magnetic field around the permanent magnet in Sec. II A, the velocity profile inside the annular
gap in Sec. II B, and the total Lorentz force acting on the magnet in Sec. II C. In Sec. III, we will
describe our experimental setup in detail (Sec. III A) and compare our analytical model with the
experimentally obtained forces (Sec. III B). Section IV summarizes the outcomes of this paper.
II. ANALYTICAL MODEL
A. Magnetic field
We consider a liquid metal confined in an annular gap bounded by an inner wall with radius
r = R and an outer wall with radius r = ηR (Fig. 1(a)); we will refer to η as the radius ratio. All walls
are treated as non-conducting and non-magnetic. The length of the pipe system is assumed to be
infinite. The geometry of the system suggests a cylindrical coordinate system with the coordinates
radius r, azimuth ϕ, and the axis of the coaxial cylinders z. The spherical permanent magnet placed
within the inner wall has the magnetic field B which will be approximated by a dipole field whose
magnetic moment m
points in axial direction. The magnetic field is not altered by the perfectly
non-conducting and non-magnetic pipe walls.
To keep the theory as simple as possible we describe the magnetic field of our permanent magnet
by that of a point-dipole which, according to Jackson,6 is given by
· r)
r
m
r ) = μ0 3 (m
.
−
B(
4π
| r |5
| r |3
(1)
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Note that (1) is also true for a circular loop of radius ρ carrying a current I. The magnitude of the
magnetic moment can then be given as6 m = π ρ 2 I.
Since Bϕ = 0 is parallel to the flow v due to the symmetry of the field and the axial component
Bz , eddy currents in the annular flow are only induced by the radial component Br . This quantity is
given by
Br =
rz
3μ0 m
· 2
.
4π
(r + z 2 )5/2
(2)
This component of the magnetic field is responsible for the induction of the eddy currents that
generate the counteracting secondary magnetic field. Note that Br changes sign at z = 0, attains its
highest magnitude at z = ±1/2r, and that it decays as zr−4 for large distances r from the dipole.
B. Velocity profile
The total Lorentz force on the magnet does not only depend on the strength of its field and the
conductivity of the material flowing past it, but it is influenced by the flow profile, as well. This
becomes clear if one considers the two extreme pipe configurations where the annular gap becomes
(a) very small, η → 1, and (b) very large, η → ∞. A velocity profile, where the majority of the
fluid passes in the vicinity of the magnet (case (a)) must cause a much higher Lorentz force than
a velocity profile, where most of the movement is far away from the magnet (case (b)). For these
reasons, we will derive the velocity field of the annular flow in this section.
Our problem is that of a steady state flow driven by a constant pressure gradient dp/dz. Under
the conditions of our experiment to be described in Sec. III, the Lorentz force acting upon the magnet
is very small (i.e., the magnet would be far from levitation if the pipe was mounted vertically and
the flow pointed upward). Hence we ignore the influence of the Lorentz forces upon the flow of
the liquid metal — an assumption that is equivalent to requiring that the Hartmann number and the
magnetic interaction parameter10, 11 of the flow are both small. In this case, the flow can be assumed
to be laminar and unidirectional and can be represented in the form v = v(r )ez . Introducing the
kinematic viscosity ν, the Navier-Stokes-equations7 simplify to
ν d
dv
dp
=
r
.
(3)
dz
r dr
dr
The velocity profile can be obtained from (3) by integration. We assume that the fluid satisfies
the no-slip boundary condition at the inner and outer wall, v(R) = v(η R) = 0. We could now express
the velocity as a function of r, R, η, and dp/dz. However, instead of the pressure gradient, we will
use the volumetric flow rate which is more practical in an actual experiment. The volumetric flow
rate Q is defined as
η R
Q = 2π R
v(r ) r dr,
2
(4)
R
and is proportional to the pressure gradient dp/dz. We thus obtain the velocity profile as a function
of the experimentally easily accessible flow rate, the inner radius, and a non-dimensional function
N(η) of the radius ratio as12
2
2
2Q
η −1 r
r
v(r ) =
N
(η)
−
1
(5)
ln
−
π R2
ln η
R
R2
with
−1
(η2 − 1)2
.
N (η) = (η2 − 1)(η2 + 1) −
ln η
(6)
The velocity profile loosely resembles that of a planar Poiseuille flow, but with the maximum
of the flow velocity shifted towards the inner pipe wall. The effect of the normalization factor
N is visualized in Fig. 2, where various velocity profiles are plotted for the same flow rate Q.
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Phys. Fluids 25, 097102 (2013)
−3
4
x 10
πR2
v
2Q
3
2
1
0
1
15
r
30
R
FIG. 2. Velocity profiles for various outer radii ηR. The flux rate and the radius of the inner pipe are the same for all curves.
The velocity profile for thin gaps is very close to parabolic, whereas it becomes more logarithmic for larger gaps. Thus, the
maximum of the velocity distribution is shifted towards the symmetry axis of the double pipe (r = 0).
The inner radius is kept fixed; the outer radius (ηR) is changed between 15 and 30. This range
is chosen because it shows best the shift from a more parabolic velocity profile for small annular
gaps to a more logarithmic profile for wider gaps. Additionally, the velocity peak moves outwards
as the outer radius of the gap is increasing. At the same time, the cross-sectional area of the
annulus is increasing, resulting in a lower mean velocity and thus a decreasing maximum velocity. If
instead of the flowrate the pressure gradient was kept constant, the velocity would increase infinitely
when η → ∞.
C. Lorentz force
We now calculate the Lorentz force acting upon the magnetic dipole. The analysis is outlined
in Sec. III A of Ref. 3. We first evaluate the eddy currents j(r ) which are induced by the flow of the
). We then determine the Lorentz force density
liquid metal through the applied magnetic field B(r
From this we obtain the total Lorentz force F by integrating over the whole fluid
f(r ) = j × B.
volume. This braking Lorentz force is evoked inside the annular flow; its counteracting force on the
magnetic dipole inside the annulus is equal in magnitude but opposite in sign to it and therefore
given by − F.
To find the eddy current density, we use Ohm’s law for moving electrical conductors in the
form6
j = σ ( E + v × B),
(7)
whereas v and B are known from Secs. II A and II B, the electric field E has yet to be determined. For
this, we take the divergence of (7) and introduce the vorticity as ω = ∇ × v. Since we consider the
steady case, the electric field strength E can be represented by the gradient of the electric potential
. Assuming charge conservation ∇ · j = 0, we obtain
− ∇ E = ∇ 2 = B · ω.
(8)
The right-hand side of Eq. (8) vanishes because the magnetic field B is always perpendicular to
ω.
Note that the efficiency of force generating could be increased if the magnet moment was placed
perpendicular to the flow. However, then axial symmetry would be lost and (8) would become
non-trivial.
Integrating (8) and using the boundary conditions ∂/∂r(r = R) = ∂/∂r(r = ηR) = 0 therefore
gives a constant electric potential that we arbitrarily set to ≡ 0. This implies that j has only an
azimuthal component
jϕ (r, z) = σ v(r )Br (r, z).
(9)
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Therefore, the only non-zero component of the Lorentz force density remains
f z = − jϕ Br = −σ v(r )Br2 (r, z).
(10)
To obtain the total Lorentz force acting on the magnetic dipole inside the annulus, (10) must be
integrated over the volume of the liquid metal. The integration over z can be carried out analytically
with the help of Ref. 8, giving a value of 5π /128. As the problem is axisymmetric, the integration
over ϕ results in a factor of 2π . Then the Lorentz force becomes
45μ20 m 2 σ
F =−
1024R 3
η
1
v(ρ)
dρ,
ρ4
(11)
where we have used the abbreviation ρ = r/R. Equation (11) is the general solution for the Lorentz
force acting on a magnet that is placed inside any flow profile. A thin slice of the flow at position
r contributes to the total force with the weighting v(r )r −4 . This means that the influence of fluid
particles decreases rapidly with their increasing distance to the magnet. The sensor of Fig. 1(b) is
thus very sensitive to velocities in its immediate vicinity.
Thess et al.3 have performed a similar analysis. However, their work is valid for the Lorentz
force flowmeter — a setup, where the magnet is placed outside a (single) circular pipe and the fluid
contributions to the force increase with v(r )r 2 . Moreover, whereas the standard problem of a falling
magnet inside a pipe uses essentially the same geometry as we do here, the pipe in the standard
problem is considered a solid. The “pipe” in our problem is liquid making the Lorentz force density
distribution more complicated.
Completing the integration over r by substituting the velocity profile (5) into (11), the total
Lorentz force is represented by
5 μ20 m 2 σ Q
· f (η)
512 π R 5
(12)
1 − η2 − η3 + η5 + 6η2 (1 − η) ln η
.
η3 (η2 + 1)(η2 − 1) ln η − η3 (η2 − 1)2
(13)
F =−
with
f (η) =
Equations (12) and (13) represent our desired result. The Lorentz force on a small magnet inside
an annular flow depends linearly on the total flowrate Q and the conductivity σ . If we assume the
magnet to be of spherical shape and constant magnetization density and introduce its size as εR, the
magnetic moment scales as m ∼ ε3 R3 . If the mean velocity is kept constant, the volume flux scales
as Q ∼ R2 and the Lorentz force scales as F ∼ ε6 R3 .
For comparison, the gravitational force the magnet itself exerts on the measurement system is
given by
FG = Mg =
4
πgρm ε3 R 3 .
3
(14)
Here, M and ρ m are the mass and the density of the magnet material, respectively. Whereas the total
Lorentz force increases with ε6 R3 , the ratio of the Lorentz force on the magnet to the gravitational
force on the system is independent of R. Consequently, a larger magnet produces a proportionally
stronger signal. Even more importantly, the force ratio is proportional to ε3 implying that the inner
pipe radius R should be as close to the magnet radius εR as possible.
The dependence of the force on the nondimensional function f(η) is shown in Fig. 5 as solid line
for η > 1. The function tends to zero as η approaches infinity. That is, forces become negligible as
the outer radius of the annulus increases. This behavior can be understood by recalling the behavior
of the velocity profiles (Fig. 2): A larger outer radius ηR at a fixed flux rate implies a smaller mean
velocity v̄, which in turn implies a smaller Lorentz force F. This shows that a measurement device
similar to that depicted in Fig. 1(b) gives best results when inserted into high velocity areas.
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Phys. Fluids 25, 097102 (2013)
For a very narrow gap, i.e., for η → 1, f(η) tends to 9/4, which is the upper limit of the curve
f(η) in Fig. 5. Therefore, the total Lorentz force on the magnet assumes the finite value
F∗ = −
45 μ20 m 2 σ Q
.
2048π R 5
(15)
The volumetric flux rate Q can be redefined in terms of the mean velocity v̄: Q = π v̄ R 2 (η2 − 1),
which is in agreement with (4). As η is only slightly larger than one, we can write η = 1 + δ/R,
where δ/R 1 is the nondimensionalized thickness of the pipe. This gives us
Q ≈ 2π v̄ Rδ.
(16)
From the combination of (16) and (15), we obtain
F∗ = −
45 μ20 m 2 σ v̄δ
1024R 4
(17)
which is the same result as when considering a falling magnet of velocity v̄ inside a solid pipe of
thickness δ/R (see, e.g., Ref. 9).
If the pipe is oriented vertically and the flow is pumped upward as described in the Introduction,
it is possible to levitate the magnet. The condition for this is that the generated Lorentz force has
the same magnitude as the gravitational force on the magnet, F = −FG . To meet this condition, the
flow rate must be
Q lev =
512π Mg R 5
5μ20 m 2 σ f (η)
(18)
inside the annular gap.
III. EXPERIMENTAL RESULTS
A. Experimental setup
Experiments are conducted on a horizontal liquid metal loop with the eutectic alloy GaInSn
which is liquid at room temperature. Its material parameters include the electrical conductivity σ =
3.46 × 106 S/m, the density ρ = 6.36 × 103 kg/m3 , and the kinematic viscosity ν = 3.4 × 10−7 m2 /s.
The flow is driven by an electromagnetic pump, and the volume flux is recorded with an inductive
flow meter.
Measurements are performed on an 80 cm long plexiglass test section with a cross sectional area
of 5 cm × 5 cm.13 We place 50 cm long plastic insets with bore holes of different diameters, such
that the flow is shaped into a pipe flow. Before the inset is a honeycomb; behind the inset is a space
of approximately 20 cm (see Fig. 3). Thus, at the rear of the inset the flow cross section abruptly
expands from circular to square. The measurement system is inserted into the flow behind the inset;
the tube containing the magnet, however, extends approximately 10 cm into the inset. Thus, the
magnet is placed downstream of a roughly 40 cm long pipe flow which continues for roughly 10 cm
beyond the magnet.
The outer diameter of the tube is 14 mm (R = 7 mm); the magnet itself is a sphere of 10 mm
diameter. The diameters of the bore holes in the inset range from 15 mm to 35 mm. These correspond
to radius ratios between η = 1.06 and η = 2.5.
The magnet is attached to a deflection element whose deformation is recorded with a strain
gauge. The connection between deflection element and magnet is a lever which allows to place
the magnet inside the flow, but keep the measurement unit outside of the metal. Magnetization
of the magnet is 0.40 Am2 , which was determined by solving Eq. (1) for the magnetization and
using the magnetic flux density of 0.63 T measured on the magnet surface. Magnet density is
7.5 g/cm3 . The measurement system resolves forces of the order of 10 μN and carries a maximum
load of 1.5 N.
The maximum volume flux for each diameter is limited by the pressure at the inlet. It ranges
from more than 0.2 l/s for the biggest diameter to as little as 0.01 l/s for the smallest diameter.
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Phys. Fluids 25, 097102 (2013)
FIG. 3. Sketch of the experimental test section comprising an inset with a circular hole inside a square duct. The fluid
entering the test section first passes a honeycomb before entering the inset. The inset is 50 cm long; the measurement probe
reaches about 10 cm into the inset from the downstream end. The whole 80 cm long test section is part of the liquid metal
loop described in Ref. 13.
Note that the gap for the smallest diameter is as thin as 0.5 mm. Flow velocities depend on both the
volume flux and the inset diameter, but generally range from a few centimeters per second to almost
10 m/s (16 mm inset).
With the above parameters we can predict the forces we will measure with the described setup.
Table I lists both the absolute predicted force calculated with Eqs. (12) and (13) and the ratio of that
force to the weight of the magnet. For illustrative purposes, Table I also predicts the forces for a
typical industrial application. Here, the parameters are based on molten steel. The flowrate for the
industrial case is chosen to match a flow velocity of 3 cm/s. Ratio η ensures that the diameter of the
pipe containing the molten steel is 20 cm. The radius R of the tube for the steel casting case must
be larger than for the laboratory case if the magnet needs to be cooled. This is only necessary if
the probe is to be built to withstand chemical reactions with the steel; in a setup where the magnet
is housed in a throw-away tube, the tube may be much smaller and the generated forces will be
significantly higher.
B. Results
The results of the laboratory experiment are shown in Fig. 4. In Fig. 4(a), the measured force is
plotted versus the volume flux for each of the six inset diameters. The symbols encoding the diameters
in Fig. 4(a) are identical to the symbols in the legend of Fig. 4(b). The latter figure shows how well
the prediction (Ftheory ) matches the actually measured forces (F): deviations between the two are
more than 33% only for the very narrow gap (η = 1.06) and the volume fluxes that correspond to
significantly less than 1 m/s (for the 35 mm). Generally, the theory slightly overpredicts the measured
values for small volume fluxes and underpredicts for the larger volume fluxes. Only the very small
gap is an exception to that rule.
TABLE I. Forces predicted by the analytical model for a typical laboratory setup and for a typical industrial application of
the magnet tube in Fig. 1(b).
Laboratory
Steel casting
σ (Sm−1 )
R (mm)
Q (m3 /s)
η
F (N)
F/FG
3.46 × 106
0.80 × 106
7
7
8.66 × 10−5
9.38 × 10−4
1.77
14.3
7.31 × 10−3
9.32 × 10−5
1.90 × 10−1
2.42 × 10−3
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Phys. Fluids 25, 097102 (2013)
2
15
theory
10
F/F
F [mN]
1.5
1
35mm
25mm
20mm
18mm
16mm
15mm
5
0.5
0
0
0.05
0.1
0.15
0.2
0
0.25
0
0.05
0.1
Q [l/s]
0.15
0.2
0.25
Q [l/s]
(a)
(b)
FIG. 4. (a) Dependence of the measured force F on the volume flux Q. (b) Ratio of the measured forces to the forces predicted
with Eqs. (12) and (13). Both forces only deviate significantly for the inset with the smallest bore hole (15 mm) and for very
small velocities such as were used for the 35 mm inset (v < 25 cm/s).
It should be clarified here that the experimental flows are mostly turbulent depending on
the actual gap size and the flow velocity, while the analytical analysis is restricted to laminar flows. This could lead to an overprediction of the forces for high volumes fluxes and narrow gaps. However, since Figure 4(b) indicates quite the opposite, the effect appears to be
negligible.
Let us now turn our attention to the dependence of the total force on the radius ratio. As presented
earlier, Fig. 5 depicts the nondimensional function f(η) for values of η > 1. The theoretical curve (solid
in Fig. 5) is obtained from Eq. (13). The experimental points are obtained as follows. The data points
in Fig. 4(a) are fitted with a straight line which is then used to interpolate each curve in Fig. 4(a) to
the volume flux Q = 0.05 l/s. The force value obtained from this interpolation is nondimensionalized
according to Eq. (12). The agreement between the theoretical and the experimental curve f(η) is
remarkable. The largest deviations are found for the smallest (−23% of the theoretical value) and
the largest inset diameter (+26%). The measurement data of the other four insets match the theory
to within 10%, with the trend of theoretical predictions becoming increasingly smaller than the
experimental data, as in Fig. 4(a).
9/4
2
f(η)
35mm
25mm
20mm
18mm
16mm
15mm
1
0
0
1
2
η
3
4
FIG. 5. The dependence of f(η), which is purely a function of the ratio of outer to inner radius. Theoretical curve (solid line)
is described by Eq. (13). Experimental data points are calculated from the interpolations in Fig. 4(a) and Eq. (12) with a
flowrate of 0.05 l/s. Left (plotting) boundary is 1, for which f(η) approaches 9/4. f(η) approaches zero when the outer radius
tends to infinity.
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Phys. Fluids 25, 097102 (2013)
IV. CONCLUSION
We analytically determined the Lorentz force acting on a small permanent magnet that is placed
inside an annular gap filled with an electrically conducting fluid. The flow is assumed to be pressuredriven and laminar in the direction parallel to the symmetry axis. The resultant force is found to
depend linearly on the total flowrate and on R3 , where R is the inner radius of the annular gap. The
magnitude of the force is monotonically decreasing for increasing outer radii. For very thin pipes,
however, the force assumes a finite value that is equal to the force caused by a thin solid pipe passing
the magnet.
We tested the theory experimentally using a metal alloy liquid at room temperature. The agreement between experimental results and theoretical predictions is remarkable given the simplicity of
our theory. The dependence of the force on the flowrate and the radii is predicted correctly not only
qualitatively, but also quantitatively to within 30%.
It was mentioned before that in the case of a vertically oriented annular pipe the permanent
magnet could be levitated in principle. Our theory, and in particular Table I allows to quantify this
prediction. In the case of the laboratory experiment, it would take a flow rate of almost half a liter
per second (4.56 × 10−4 m3 /s) through a 5.5 mm wide gap (25 mm inset) to levitate the 10 mm
diameter magnet sphere. The mean upward velocity would then be around 0.7 m/s, which is more
than five times higher than the velocities we could reach with our horizontal setup.
ACKNOWLEDGMENTS
The authors would like to thank Ilko Rahneberg for the help with the design of the experiment
and Igor Sokolov and Aidin Asgharzadeh for the help performing the measurements. The work was
funded by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the Research Training
Group “Lorentz Force Velocimetry and Lorentz Force Eddy Current Testing.”
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